Pasha Zusmanovich - teaching

A.T. Fomenko, Double cover of the Klein bottle by torus

Pasha Zusmanovich

6/7TOPO Topology, winter semester 2024/2025

Teams

LITERATURE
Main:

J.R. Munkres, Topology, 2nd ed., Prentice Hall, 2000
O.Ya. Viro, O.A. Ivanov, N.Yu. Netsvetaev, V.M. Kharlamov, Elementary Topology. Problem Textbook, AMS, 2008 Referred in what follows as [V]

Additional:

A.V. Arkhangelskii and V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, D. Reidel, 1984
R. Engelking, General Topology, revised and completed ed., Heldermann Verlag, 1989
A. Fomenko and D. Fuchs, Homotopical Topology, 2nd ed., Springer, 2016 (included here solely for its pictures)
M.W. Hirsch, Differential Topology, Springer, 2001
J.F. Kelley, General Topology, Van Nostrand, 1955; reprinted by Springer, 1975 and Dover, 2017
C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, 1980 (Russian edition only)
J.W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965
S.A. Morris, Topology Without Tears
M. Starbird and F. Su, Topology Through Inquiry, MAA Press, 2019

All the books are available in electronic form in multiple places.

= available in the university library

A BRIEF SYNOPSIS

Class 1 September 24, 2024
Organizational details. The subject of topology. Definition of a topological space. Examples of topological spaces: trivial and discrete topology, topologies on 0-, 1- and 2-element sets, the standard topology on \(\mathbb R\). Number of topologies on a finite set. Finer and coarser topologies. Closed sets. The closed interval in \(\mathbb R\) is closed in the standard topology. Clopen sets. A parody of an episode from the movie "The Bunker" featuring clopen sets.
([V], pp. xi-xii, 11-14; Munkres, p.76).

Class 2 October 1, 2024
More examples: topologies on finite sets, the arrow. Cantor set as an example of a closed set. Base of a topology, examples. Any base of the standard topology on \(\mathbb R\) can be decreased.
([V], pp.11-12,16).

Class 3 October 8, 2024
Discussion of homeworks 1,2. Criteria for a set to be a base ([V],p.16,3.A,3.B,3.C). Criterion for a topology to be coarser (Munkres, Lemma 13.3).The standard topology on \(\mathbb R^2\).
([V], pp.16-17).

Class 4 October 15, 2024
Discussion of homeworks 3,4. Metric spaces, 0-1 metric, Euclidean metric, Manhattan metric, metrics on finite sets. Restriction of a metric to a subspace. Balls and spheres. Metric topology.
([V], pp.18-20,22).

Class 5 October 22, 2024
Discussion of homeworks 5,6. Metrizability of topological spaces, examples of metrizable and non-metrizable spaces. Operations on metrics. Separation property. Metric topology always satisfies the separation property. Topological equivalence and metric equivalence of metrics. If \(\rho\) is a metric, then \(\frac{\rho}{1+\rho}\) is a metric.
([V], pp.22-23; Munkres, pp.119-122).

Class 6 October 29, 2024
Metric equivalence implies topological equivalence. If \(\rho\) is a metric, then \(\frac{\rho}{1+\rho}\) is a topologically equivalent, but not necessary metrically equivalent, metric. Subspace topology, examples. Interior.
([V], pp.23,27-29; Munkres, pp.88-90).

video (partially)

Class 7 November 5, 2024
Discussion of homeworks 7-10. Closure, boundary. Definition of continuous and open (i.e., an image of an open set is open) maps. \(x \mapsto x^2\) is continuous but not open. Equivalent conditions for continuity.
([V], pp.30-31,59; Munkres, pp.102-103). video

Class 8 November 12, 2024
Discussion of homeworks 11-13. The continuity of a map \(f: X \to Y\) is equivalent to each of the following conditions: \[ f^{-1}(Cl(B)) \supseteq Cl(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f^{-1}(Int(B)) \subseteq Int(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f(Cl(A)) \subseteq Cl(f(A)) \text{ for any } A \subseteq X \\ f(Int(A)) \supseteq Int(f(A)) \text{ for any } A \subseteq X . \] Further examples and non-examples of continuous maps. Continuity at a point. Equivalence with the \(\epsilon-\delta\) definiton of continuity from analysis in the case of metric topologies. Homeomorphisms (definition, elementary properties).
([V], pp.60-61,67; Munkres, pp.102-105). video

Class 9 November 19, 2024
Discussion of homeworks 14-16. Homeomorphisms of a topological space to itself form a group. Examples of homeomorphisms: subspace topologies on various sets in \(\mathbb R\) and \(\mathbb R^2\). Planes with punctures are homeomorphic (in the subspace topology of the standard topology) iff the cardinalities of the sets of punctured points are the same (done partially).
([V], pp.69-70,72; Munkres, pp.105-106). video

Class 10 November 26, 2024
Connectedness, examples of connected, disconnected, and totally disconnceted topological spaces, connected subsets, connected components.
([V], pp.83-86). video

Class 11 December 3, 2024
Discussion of homeworks 17-21. Connected subsets of \(\mathbb R\) with the standard topology are exactly all kind of intervals (open, closed, semiopen, semiinfinite), singletons, and the whole \(\mathbb R\). Topological generalization of the Intermediate Value Theorem from basic analysis: an image of a continuous map from a connected topological space to \(\mathbb R\) with the standard topology, is an interval. Using connectedness to prove non-homeomorphism of topological spaces.
([V], pp.87,89). video (sluggish, barely usable)

Class 12 December 10, 2024
Discussion of homeworks 22-23. Compact space, compact subspace: definition, examples and non-examples. Heine-Borel theorem in topological setting. Compactness is preserved under continuous maps.
([V], pp.108-111). video (sluggish, barely usable)

Class 13 December 17, 2024
Discussion of homeworks 24-27. Product topology. Quotient topology.
([V], pp.136-137,142,145-148).

HOMEWORKS

EXAM

Set 1:

Set 3:

Set 4:

A brief synopsis, videos, and homeworks from this course at the previous semesters

Created: Fri Oct 2 2020
Last modified: Wed Dec 18 2024 17:26:18 CET